A theorem of validity for Edgeworth expansions. (English) Zbl 0595.62119

Sargan, J. D.; Satchell, S. E.

Econometrica 54, 189-213 (1986).

A method for approximating the exact densities of the estimators by Edgeworth expansion is derived and applied to an autoregressive equation which frequently arises in economic models. Let c(p,T) be a vector of statistics where p denotes the sample moments and T the sample size. We need to derive the exact distribution function of \[ g(x)=\sqrt{T}(c(p,T)-c(\mu,T)), \] where \(\mu\) is the expectation of p. Since it is impossible in general to derive the exact distribution, the authors propose its Edgeworth approximation, which is proved under certain stated regularity and continuity conditions to have an error of order \(o(T^{-r/2})\), where the Edgeworth approximation is of order r. This is applied to a linear dynamic equation \[ y_ t=\sum^{s}_{j=1}a_ jy_{t-j}+\sum^{K}_{j=1}b_ jx_{tj}+u_ t \] where \(y_ t\) is an endogenous variable, \(y_{t-j}\) are lagged values of \(y_ t\), \(x_{tj}\) are values of K exogenous variables and \(u_ t\) is an error process with mean zero and variance \(\sigma^ 2\).

Reviewer: J.K.Sengupta

MSC:

62P20	Applications of statistics to economics
62E20	Asymptotic distribution theory in statistics
62H10	Multivariate distribution of statistics

Keywords:

autoregressive model; asymptotic approximation of density; functions; simultaneous equation models; Edgeworth expansion